Optimization, Stability and Cylindrical Decomposition

Some connections between constructive real algebraic geometry and constrained optimization are exploited. We show how the problem of determining the projection of a real-algebraic variety on a certain axis is equivalent to a problem in nonlinear programming. As an application, Grr obner bases are used to deal with an optimization problem arising in the theory of local Lyapunov functions. The problems addressed are: determining critical levels of local Lyapunov functions and investigating robustness using Lyapunov functions. Since the tools used come from commutative algebra and algebraic geometry the diierential equations considered are of polynomial type and the Lyapunov functions used are polynomial.